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I don't know yet how to integrate this function...

  1. Sep 29, 2015 #1

    pabilbado

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    I was just wondering about the path an orbiting particle takes when it's center of circular motion also translates. So as to calculate the distance traveled I know I have to get a function that relates velocity with respect to time and integrate it, but I can't doit, I get stuck integrating. I have search the Internet across and nothing. Possibly I don't know what I am looking for. Here is the function:
    df40425500452af6d860f2e1c2e4efdb.png
    I can simplify it but it continues being difficult: ba863ee3701d465bae822f56ed19c330.png
     
  2. jcsd
  3. Sep 29, 2015 #2
    Is this a homework problem?
     
  4. Sep 30, 2015 #3

    pabilbado

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    No, is not. Two days ago it was my father's birthday (he is a nerdy engineer so he will like this kind of present) so I am trying to calculate how much kilometers he has travelled around the sun. And as an approximation I have made the earth's orbit a straight line, and this is the result of a point on earth rotating and translating.
     
  5. Sep 30, 2015 #4

    SteamKing

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    I don't know how you can make an orbit a straight line.

    If you want to know how far your father has traveled in his lifetime, assume that earth's orbit is a circle with a radius of 150 million kilometers. Every year, the earth makes one orbit of the sun, by definition. The eccentricity of earth's orbit is about 0.017, so a circle is a good approximation for all practical purposes here.
     
  6. Sep 30, 2015 #5

    pabilbado

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    Yeah, that's the easy way. What I want is to take into account the rotation of earth, while is moving around the sun. Because a point in the surface of the earth does not travel a circular path around the sun but rather a really "curly" trajectory. So as a first approximation I have flattened earth's orbit so it is just traveling in a straight line and I need to take into consideration the that the point is also rotating. That wwhat the integral is for.
     
  7. Sep 30, 2015 #6
    Why not write the parametric equations as:

    x(t) = Ra*cos(2*pi t/Ta) + Rb*cos(2*pi*t/Tb)*cos(latitude)
    y(t) = Ra*sin(2*pi t/Ta) + Rb*sin(2*pi*t/Tb)*cos(latitude)

    Use the arc length integral, substitute in, and integrate numerically.

    Ra, Ta = radius and period of earth's orbit
    Rb = earth's radius, Tb = one day

    Hopefully, you can treat the latitude as constant.
     
  8. Sep 30, 2015 #7

    pabilbado

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    EDIT: thinking about it I believe I will be stuck with the same problem: integrating a function that has a trigonometric function inside a square root.

    Yeah I thought about it, but I had two different directions, but I wasn't sure if I could just use the arc lenght for the first equation, then for the second one and sum them over. Is that what I have to do? If it wasn't the case just to satisfy my curiosity how could I integrate the first function I posted?(My approximate scenario would also yield a function for the movement of a particle around a translating and rotating sphere, wouldn't it?).
     
  9. Dec 5, 2015 #8

    pabilbado

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    I forgot to post how I solved the problem. I did the taylor series and then I integrated term by term. I did it a long time ago, but i forgott to post it here.
     
  10. Feb 18, 2016 #9

    Fervent Freyja

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    Something so thoughtful and sweet should have a place on the refrigerator forever! :smile:
     
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